Newtonian dynamics 7 You might think that the strangeness of contracting flows, flows such as the Rössler flow of Fig. 2.4 is of concern only to chemists; real physicists do Hamiltonians, right? Not at all - while it is easier to visualize aperiodic dynamics when a flow is contracting onto a lower-dimensional attracting set, there are plenty examples of chaotic flows that do preserve the full symplectic invariance of Hamiltonian dynamics. The truth is, the whole story started with Poincaré s restricted 3-body problem, a realization that chaos rules also in general (non-hamiltonian) flows came much later. Here we briefly review parts of classical dynamics that we will need later on; symplectic invariance, canonical transformations, and stability of Hamiltonian flows. We discuss billiard dynamics in some detail in Chapter 8. 7.1 Hamiltonian flows 89 7.2 Stability of Hamiltonian flows 90 7.3 Symplectic maps 93 Further reading 97 Exercises 97 References 98 7.1 Hamiltonian flows (P. Cvitanović and L.V. Vela-Arevalo) An important class of flows are Hamiltonian flows, given by a Hamiltonian H(q, p) together with the Hamilton s equations of motion Appendix 20 q i = H, p i p i = H, (7.1) q i with the 2D phase space coordinates x split into the configuration space coordinates and the conjugate momenta of a Hamiltonian system with D degrees of freedom (dof): x =(q, p), q =(q 1,q 2,...,q D ), p =(p 1,p 2,...,p D ). (7.2) Section?? The energy, or the value of the Hamiltonian function at the state space point x =(q, p) is constant along the trajectory x(t), d H H(q(t), p(t)) = q i (t)+ H ṗ i (t) dt q i p i = H H H H =0, (7.3) q i p i p i q i so the trajectories lie on surfaces of constant energy, or level sets of the Hamiltonian {(q, p) :H(q, p) =E}. For 1-dof Hamiltonian systems this is basically the whole story.
90 CHAPTER 7. NEWTONIAN DYNAMICS Example 7.1 Unforced undamped Duffing oscillator: When the damping term is removed from the Duffing oscillator (2.7), the system can be written in Hamiltonian form with the Hamiltonian 1 0 p q H(q,p) = p2 2 q2 2 + q4 4. (7.4) This is a 1-dof Hamiltonian system, with a 2-dimensional state space, the plane (q, p). The Hamilton s equations (7.1) are q = p, ṗ = q q 3. (7.5) For 1-dof systems, the surfaces of constant energy (7.3) are simply curves in the phase plane (q, p), and the dynamics is very simple: the curves of constant energy are the trajectories, as shown in Fig. 7.1. Thus all 1-dof systems are integrable, in the sense that the entire phase plane is foliated by curves of constant energy, either periodic as is the case for the harmonic oscillator (a bound state ) or open (a scattering trajectory ). Add one more degree of freedom, and chaos breaks loose. 1 2 1 0 1 2 Fig. 7.1 Phase plane of the unforced, undamped Duffing oscillator. The trajectories lie on level sets of the Hamiltonian (7.4). 10 Example 6.1 Chapter?? Example 7.2 Collinear helium: In Chapter??, we shall apply the periodic orbit theory to the quantization of helium. In particular, we will study collinear helium, a doubly charged nucleus with two electrons arranged on a line, an electron on each side of the nucleus. The Hamiltonian for this system is H = 1 2 p2 1 + 1 2 p2 2 2 r 1 2 r 2 + 1 r 1 + r 2. (7.6) Collinear helium has 2 dof, and thus a 4-dimensional phase space M, which energy conservation reduces to 3 dimensions. The dynamics can be projected onto the 2-dimensional configuration plane, the (r 1,r 2), r i 0 quadrant, Fig. 7.2. It looks messy, and, indeed, it will turn out to be no less chaotic than a pinball bouncing between three disks. As always, a Poincaré section will be more informative than this rather arbitrary projection of the flow. r 2 8 6 4 2 0 0 2 4 6 8 10 r 1 Note an important property of Hamiltonian flows: if the Hamilton equations (7.1) are rewritten in the 2D phase space form ẋ i = v i (ssp), the divergence of the velocity field v vanishes, namely the flow is incompressible. The symplectic invariance requirements are actually more stringent than just the phase space volume conservation, as we shall see in the next section. 7.2 Stability of Hamiltonian flows Fig. 7.2 A typical collinear helium trajectory in the [r 1,r 2 ] plane; the trajectory enters along the r 1 -axis and then, like almost every other trajectory, after a few bounces escapes to infinity, in this case along the r 2 -axis. Hamiltonian flows offer an illustration of the ways in which an invariance of equations of motion can affect the dynamics. In the case at hand, the symplectic invariance will reduce the number of independent stability eigenvalues by a factor of 2 or 4. newton - 25sep2007 ChaosBook.org version11.9.2, Aug 21 2007
7.2. STABILITY OF HAMILTONIAN FLOWS 91 7.2.1 Canonical transformations The equations of motion for a time-independent, D-dof Hamiltonian (7.1) can be written ( ) 0 I ẋ i = ω ij H j (x), ω =, H I 0 j (x) = H(x), (7.7) x j where x =(q, p) Mis a phase space point, H k = k H is the column vector of partial derivatives of H, I is the [D D] unit matrix, and ω the [2D 2D] symplectic form ω T = ω, ω 2 = 1. (7.8) To stress the peculiar properties of Hamiltonian flows, we change the notation slightly, and denote the fundamental matrix (4.6) by M t (x). The evolution of M t is again determined by the stability matrix A, (4.9): d dt M t (x) =A(x)M t (x), A ij (x) =ω ik H kj (x), (7.9) where the matrix of second derivatives H kn = k n H is called the Hessian matrix. From the symmetry of H kn it follows that A T ω + ωa =0. (7.10) This is the defining property for infinitesimal generators of symplectic (or canonical) transformations, transformations which leave the symplectic form ω invariant. Symplectic matrices are by definition linear transformations that leave the (antisymmetric) quadratic form x i ω ij y j invariant. This immediately implies that any symplectic matrix satisfies Q T ωq = ω, (7.11) and when Q is close to the identity Q = 1 + δta it follows that that A must satisfy (7.10). In group language this means that the property (7.11) defines the symplectic group Sp(2D), just as the Lie group of orthogonal matrices O(d) is defined by linear transformations that preserve the (symmetric) quadratic form x 2 = x i δ ij x j, The symplectic Lie algebra sp(2d) follows by writing Q =exp(δta) and linearizing Q =1+δtA. This yields (7.10) as the defining property of infinitesimal symplectic transformations. Consider now a smooth nonlinear change of variables of form y i = h i (x), and define a new function K(x) =H(h(x)). Under which conditions does K generate a Hamiltonian flow? In what follows we will use the notation j = / y j : by employing the chain rule we have that 7.2, page 97 ChaosBook.org version11.9.2, Aug 21 2007 ω ij j K = ω ij h l x j l H (7.12) newton - 25sep2007
92 CHAPTER 7. NEWTONIAN DYNAMICS complex saddle Example 6.1 saddle center By virtue of (7.1) l H = ω lm ẏ m, so that, again by employing the chain rule, we obtain h l h m ω ij j K = ω ij ω lm ẋ n (7.13) x j x n The right hand side simplifies to ẋ i (yielding Hamiltonian structure) only if h l h m ω ij ω lm = δ in (7.14) x j x n or, in compact notation, by defining ( h) ij = hi x j ω( h) T ω( h) =1 (7.15) which is equivalent to the requirement that h is symplectic. h is then called a canonical transformation. We care about canonical transformations for two reasons. First (and this is a dark art), if the canonical transformation h is very cleverly chosen, the flow in new coordinates might be considerably simpler than the original flow. Second, Hamiltonian flows themselves are a prime example of canonical transformations. Example 7.3 Hamiltonian flows are canonical: ( For Hamiltonian flows it follows from (7.10) that d dt M T ωm ) =0, and since at the initial time M 0 (x 0)=1, M is a symplectic transformation (7.11). This equality is valid for all times, so a Hamiltonian flow f t (x) is a canonical transformation, with the linearization xf t (x) a symplectic transformation (7.11): For notational brevity here we have suppressed the dependence on time and the initial point, M = M t (x 0). By elementary properties of determinants it follows from (7.11) that Hamiltonian flows are phase space volume preserving: det M =1. (7.16) Actually it turns out that for symplectic matrices (on any field) one always has det M =+1. degenerate saddle generic center real saddle degenerate center Fig. 7.3 Stability exponents of a Hamiltonian equilibrium point, 2-dof. 7.4, page 98 Section 4.3.1 7.5, page 98 7.2.2 Stability of equilibria of Hamiltonian flows For an equilibrium point x q the stability matrix A is constant. Its eigenvalues describe the linear stability of the equilibrium point. In the case of Hamiltonian flows, from (7.10) it follows that the characteristic polynomial of A for an equilibrium x q satisfies det (A λ1) = det (ω 1 (A λ1)ω) =det ( ωaω λ1) = det (A T + λ1) = det (A + λ1). (7.17) A is the matrix (7.10) with real matrix elements, so its eigenvalues (the stability exponents of (4.27)) are either real or come in complex pairs. Symplectic invariance implies in addition that if λ is an eigenvalue, then λ, λ and λ are also eigenvalues. Distinct symmetry classes of the stability exponents of an equilibrium point in a 2-dof system are displayed in Fig. 7.3. It is worth noting that while the linear stability of equilibria in a Hamiltonian system always respects this symmetry, the nonlinear stability can be completely different. newton - 25sep2007 ChaosBook.org version11.9.2, Aug 21 2007
7.3. SYMPLECTIC MAPS 93 7.3 Symplectic maps A stability eigenvalue Λ=Λ(x 0,t) associated to a trajectory is an eigenvalue of the monodromy matrix M. AsM is symplectic, (7.11) implies that M 1 = ωm T ω, (7.18) so the characteristic polynomial is reflexive, namely it satisfies det (M Λ1) = det (M T Λ1) =det ( ωm T ω Λ1) = det (M 1 Λ1) =det (M 1 ) det (1 ΛM) = Λ 2D det (M Λ 1 1). (7.19) Hence if Λ is an eigenvalue of M, soare1/λ, Λ and 1/Λ. Real (non-marginal, Λ 1) eigenvalues always come paired as Λ, 1/Λ. The Liouville conservation of phase space volumes (7.16) is an immediate consequence of this pairing up of eigenvalues. The complex eigenvalues come in pairs Λ, Λ, Λ =1, or in loxodromic quartets Λ, 1/Λ, Λ and 1/Λ. These possibilities are illustrated in Fig. 7.4. Example 7.4 Hamiltonian Hénon map, reversibility: By (4.41) the Hénon map (3.15) for b = 1 value is the simplest 2-d orientation preserving area-preserving map, often studied to better understand topology and symmetries of Poincaré sections of 2 dof Hamiltonian flows. We find it convenient to multiply (3.16) by a and absorb the a factor into x in order to bring the Hénon map for the b = 1 parameter value into the form x i+1 + x i 1 = a x 2 i, i =1,..., n p, (7.20) The 2-dimensional Hénon map for b = 1 parameter value complex saddle degenerate saddle generic center saddle center real saddle degenerate center x n+1 = a x 2 n y n y n+1 = x n. (7.21) is Hamiltonian (symplectic) in the sense that it preserves area in the [x, y] plane. For definitiveness, in numerical calculations in examples to follow we shall fix (arbitrarily) the stretching parameter value to a =6, a value large enough to guarantee that all roots of 0=f n (x) x (periodic points) are real. Fig. 7.4 Stability of a symplectic map in R 4. 8.6, page 104 Example 7.5 2-dimensional symplectic maps: In the 2-dimensional case the eigenvalues (5.2) depend only on tr M t Λ 1,2 = 1 2 ( tr M t ± ) (tr M t 2)(tr M t +2). (7.22) The trajectory is elliptic if the stability residue tr M t 2 0, with complex eigenvalues Λ 1 = e iθt, Λ 2 =Λ 1 = e iθt.if tr M t 2 > 0, λ is real, and the trajectory is either hyperbolic Λ 1 = e λt, Λ 2 = e λt, or (7.23) inverse hyperbolic Λ 1 = e λt, Λ 2 = e λt. (7.24) ChaosBook.org version11.9.2, Aug 21 2007 newton - 25sep2007
94 CHAPTER 7. NEWTONIAN DYNAMICS 7.3.1 The standard map Truth is rarely pure, and never simple. Oscar Wilde Given a smooth function g(x), the map x n+1 = x n + y n+1 y n+1 = y n + g(x n ) (7.25) is an area-preserving map. The corresponding monodromy matrix is ( ) 1+g M(x, y) = (x) 1 g (7.26) (x) 1 det M =1so the map preserves areas, moreover one can easily check that M is symplectic. In particular one can consider x as an angle, and y as the conjugate angular momentum, with a function g periodic, with period 2π. The phase space of the map is thus the cylinder S 1 R (S 1 being the 1-torus): by taking (7.25) mod 2π the map can be reduced on the 2-torus S 2. Note that the mapping provides a stroboscopic view of the flow generated by a pulsed Hamiltonian H(x, y; t) = 1 2 y2 + G(x)δ 1 (t) (7.27) where δ 1 denotes the periodic delta function δ 1 (t) = m= δ(t m) (7.28) and G (x) = g(x). (7.29) The standard map corresponds to the choice g(x) =k sin(x): the corresponding map will be denoted by A. When k =0angular momentum is conserved, and orbits are pure rotations, motion is periodic or quasiperiodic according to y 0 being rational or irrational; invariant tori are straight lines in the (x, y) phase plane. Despite the simple structure of the standard mapping, a complete description of its dynamics for arbitrary values of the nonlinear parameter k is fairly complex: small k regime falls within KAM scheme, yet any perturbative regime fails very soon as k increases. It turns out that interesting features of this map, including transition to global chaos (destruction of the last deformed invariant torus), may be tackled by detailed investigation of the stability of periodic orbits. A compact index of stability of a Q-periodic orbit is provided by the residue: R Q = 1 ( 2 tr M Q ) ; (7.30) 4 newton - 25sep2007 ChaosBook.org version11.9.2, Aug 21 2007
7.3. SYMPLECTIC MAPS 95 as a matter of fact if R Q (0, 1) the orbit is elliptic, R Q > 1 corresponds to hyperbolic orbits and, finally, R Q < 0 marks the case of inverse hyperbolic orbits. For k =0all orbits with y 0 = P/Q are periodic with period Q (and winding number P/Q): since x 0 is arbitrary, actually they are organized in families (and they all have R Q =0). As soon as k increases there is only a finite number of such orbits that survive, according to Poincaré- Birkhoff theorem, half of them are elliptic, and half hyperbolic. If we further vary k in such a way that the residue of the elliptic Q cycle go to 1 a bifurcation takes place, and two or more periodic orbits of higher period are generated. In practice the search for remarkable classes of periodic orbits for the standard map takes advantage of an important symmetry property: A can be written as the product of two involutions T 1 and T 2 (involution means that the square of the map is the identity): A = T 2 T 1 (7.31) where T 1 ( x y ) = ( x y k sin x ) (7.32) and T 2 ( x y ) = ( x + y y ). (7.33) Now define symmetry lines L 1 and L 2 as the set of fixed points of the corresponding involution: L 1 consists of the lines x =0,π, L 2 of x = y/2 mod (2π). There are deep connections between symmetry lines and periodic orbits: we just give an example with the following statement: if (x 0,y 0 ) L 1 and A M (x 0,y 0 ) L 1 (i.e. they are both fixed points of T 1 ), then (x 0,y 0 ) is a periodic point of period 2M. As a matter of fact A 2M (x 0,y 0 ) = A M 1 T 2 T 1 A M 1 T 2 T 1 (x 0,y 0 ) = A M 1 T 2 A M 1 T 2 (x 0,y 0 ) (7.34) by the fixed point property. Now the involution property implies T 2 A = T 1 AT 1 = T 2 (7.35) and thus and AT 2 AT 2 = AT 1 T 2 = 1 (7.36) A P T 2 A P T 2 = A P 1 T 2 A P 1 T 2 (7.37) from which it easily follows that (x 0,y 0 ) belongs to a 2M cycle. 7.3.2 Poincaré invariants Let C a region in the phase space and V (0) its volume. Denoting the flow of the Hamiltonian system by f t (x), the volume of C after a time ChaosBook.org version11.9.2, Aug 21 2007 newton - 25sep2007
96 CHAPTER 7. NEWTONIAN DYNAMICS t is V (t) =f t (C), and using (7.16) we derive the Liouville theorem: V (t) = dx = f t (x ) f t (C) C x dx det (M)dx = dx = V (0), (7.38) C Hamiltonian flows preserve phase space volumes. The symplectic structure of Hamilton s equations buys us much more than the incompressibility, or the phase space volume conservation. Consider the symplectic product of two infinitesimal vectors (δx, δˆx) = δx T ωδˆx = δp i δˆq i δq i δˆp i D = {oriented area in the (q i,p i ) plane}. (7.39) Time t later we have i=1 (δx,δˆx )=δx T M T ωmδˆx = δx T ωδˆx. This has the following geometrical meaning. We imagine there is a reference phase space point. We then define two other points infinitesimally close so that the vectors δx and δˆx describe their displacements relative to the reference point. Under the dynamics, the three points are mapped to three new points which are still infinitesimally close to one another. The meaning of the above expression is that the area of the parallelopiped spanned by the three final points is the same as that spanned by the inital points. The integral (Stokes theorem) version of this infinitesimal area invariance states that for Hamiltonian flows the D oriented areas V i bounded by D loops ΩV i, one per each (q i,p i ) plane, are separately conserved: dp dq = p dq = invariant. (7.40) V ΩV Morally a Hamiltonian flow is really D-dimensional, even though its phase space is 2D-dimensional. Hence for Hamiltonian flows one emphasizes D, the number of the degrees of freedom. C in depth: Appendix B.1, p. 291 In theory there is no difference between theory and practice. In practice there is. Yogi Berra newton - 25sep2007 ChaosBook.org version11.9.2, Aug 21 2007
Further reading 97 Further reading Hamiltonian dynamics literature. If you are reading this book, in theory you already know everything that is in this chapter. In practice you do not. Try this: Put your right hand on your heart and say: I understand why nature prefers symplectic geometry. Honest? We make an attempt in Section??. Out there there are about 2 centuries of accumulated literature on Hamilton, Lagrange, Jacobi etc. formulation of mechanics, some of it excellent. In context of what we will need here, we make a very subjective recommendation we enjoyed reading Percival and Richards [10] and Ozorio de Almeida [11]. Symplectic. The term symplectic Greek for twining or plaiting together was introduced into mathematics by Hermann Weyl. Canonical lineage is church-doctrinal: Greek kanon, referring to a reed used for measurement, came to mean in Latin a rule or a standard. The sign convention of ω. The overall sign of ω, the symplectic invariant in (7.7), is set by the convention that the Hamilton s principal function (for energy conserving flows) is given by R(q,q,t)= q pidqi Et. With this q sign convention the action along a classical path is minimal, and the kinetic energy of a free particle is positive. Symmetries of the symbol square. For a more detailed discussion of symmetry lines see Refs. [4, 8, 49, 9]. Standard map. Standard maps have been very extensively studied (also in their quantum counterpart): as a starting point see the reviews Refs. [10, 11]. Exercises (7.1) Complex nonlinear Schrödinger equation. Consider the complex nonlinear Schrödinger equation in one spatial dimension [1]: i φ t + 2 φ x + 2 βφ φ 2 =0, β 0. (a) Show that the function ψ : R C defining the traveling wave solution φ(x, t) =ψ(x ct) for c>0satisfies a second-order complex differential equation equivalent to a Hamiltonian system in R 4 relative to the noncanonical symplectic form whose matrix is given by w c = 0 0 1 0 0 0 0 1 1 0 0 c 0 1 c 0. (b) Analyze the equilibria of the resulting Hamiltonian system in R 4 and determine their linear stability properties. (c) Let ψ(s) =e ics/2 a(s) for a real function a(s) and determine a second order equation for a(s). Show that the resulting equation is Hamiltonian and has heteroclinic orbits for β<0. Find them. ChaosBook.org version11.9.2, Aug 21 2007 (d) Find soliton solutions for the complex nonlinear Schrödinger equation. (Luz V. Vela-Arevalo) (7.2) Symplectic group/algebra Show that if a matrix C satisfies (7.10), then exp(sc) is a symplectic matrix. (7.3) When is a linear transformation canonical? (a) Let A be a [n n] invertible matrix. Show that the map φ : R 2n R 2n given by (q, p) (Aq, (A 1 ) T p) is a canonical transformation. (b) If R is a rotation in R 3, show that the map (q, p) (Rq, Rp) is a canonical transformation. (Luz V. Vela-Arevalo) (7.4) Determinant of symplectic matrices. Show that the determinant of a symplectic matrix is +1, by going through the following steps: (a) use (7.19) to prove that for eigenvalue pairs each member has the same multiplicity (the same holds for quartet members), (b) prove that the joint multiplicity of λ = ±1 is even, exernewton - 13jun2004
98 Exercises (c) show that the multiplicities of λ = 1 and λ = 1 cannot be both odd. (Hint: write P (λ) =(λ 1) 2m+1 (λ +1) 2l+1 Q(λ) and show that Q(1) = 0). (7.5) Cherry s example. What follows Refs. [2, 3] is mostly a reading exercise, about a Hamiltonian system that is linearly stable but nonlinearly unstable. Consider the Hamiltonian system on R 4 given by H = 1 2 (q2 1 + p 2 1) (q 2 2 + p 2 2)+ 1 2 p2(p2 1 q 2 1) q 1q 2p 1. (a) Show that this system has an equilibrium at the origin, which is linearly stable. (The linearized system consists of two uncoupled oscillators with frequencies in ratios 2:1). (b) Convince yourself that the following is a family of solutions parametrized by a constant τ: q 1 = cos(t τ) cos 2(t τ) 2, q 2 =, t τ t τ p 1 = sin(t τ) sin 2(t τ) 2, p 2 =. t τ t τ These solutions clearly blow up in finite time; however they start at t =0at a distance 3/τ from the origin, so by choosing τ large, we can find solutions starting arbitrarily close to the origin, yet going to infinity in a finite time, so the origin is nonlinearly unstable. (Luz V. Vela-Arevalo) References [1] J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry (Springer, New York, 1994). [2] T.M. Cherry, Some examples of trajectories defined by differential equations of a generalized dynamical type, Trans.Camb.Phil.Soc. XXIII, 165 (1925). [3] K.R. Meyer, Counter-examples in dynamical systems via normal form theory, SIAM Review 28, 41 (1986) [4] D.G. Sterling, H.R. Dullin and J.D. Meiss, Homoclinic Bifurcations for the Hénon Map, Physica D 134, 153 (1999); chao-dyn/9904019. [5] H.R. Dullin, J.D. Meiss and D.G. Sterling, Symbolic Codes for Rotational Orbits, nlin.cd/0408015. [6] A. Gómez and J.D. Meiss, Reversible Polynomial Automorphisms of the Plane: the Involutory Case, Phys. Lett. A 312, 49 (2003); nlin.cd/0209055. [7] J.M. Greene, A method for determining a stochastic transition, J. Math. Phys. 20, 1183 (1979). [8] J.M. Greene, Two-Dimensional Measure-Preserving Mappings, J. Math. Phys. 9, 760 (1968) [9] S.J. Shenker and L.P. Kadanoff, Critical behavior of a KAM surface: I. Empirical results, J.Stat.Phys. 27, 631 (1982). [10] B.V. Chirikov, A Universal Instability of Many-Dimensional Oscillator System, Phys.Rep. 52, 265 (1979). [11] J.D. Meiss, Symplectic maps, variational principles, and transport, Rev.Mod.Phys. 64, 795 (1992). refsnewt - 7aug2005 ChaosBook.org version11.9.2, Aug 21 2007